Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").

For n > 1, a(n) is the number of functions f:{1,2}->{1,...,n+2} where f(1) > 1 and f(2) > 2. Note that there are n+1 possible values for f(1) and n possible values for f(2). For example, a(3) = 12 since there are 12 functions f from {1,2} to {1,2,3,4,5} with f(1) > 1 and f(2) > 2. - Dennis P. Walsh, Dec 24 2011

a(m), for m>=1, are the only positive integer values t for which the Binet-de Moivre formula for the recurrence b(n)=b(n-1)+t*b(n-2) with b(0)=0 and b(1)=1 has a root of a square. PROOF (as suggested by Wolfdieter Lang, Mar 26 2014): The sqrt(1+4t) appearing in the zeros r1 and r2 of the characteristic equation is (a positive) integer for positive integer t precisely if 4t+1=(2m+1)^2, that is t=a(m), m>=1. Thus, the characteristic roots are integers: r1=m+1 and r2=-m.

a(n) is the maximum number of queens of one color that can coexist without attacking one queen of the opponent's color on an [n+2] X [n+2] chessboard. The lone queen can be placed in any position on the perimeter of the board. - Bob Selcoe, Feb 07 2015

With a(0) = 1, a(n-1) is the smallest positive number not in the sequence such that Sum_{i=1..n} 1/a(i-1) has a denominator equal to n. - Derek Orr, Jun 17 2015

The positive members of this sequence are a proper subsequence of the so-called 1-happy couple products A007969. See the W. Lang link there, eq. (4), with Y_0 = 1, with a table at the end. - Wolfdieter Lang, Sep 19 2015

For n > 0, a(n) is the reciprocal of the area bounded above by y = x^(n-1) and below by y = x^n for x in the interval [0,1]. Summing all such areas visually demonstrates the formula below giving Sum_{n>=1} 1/a(n) = 1. - Rick L. Shepherd, Oct 26 2015

It appears that, except for a(0)=0, this is the set of positive integers n such that x*floor(x) = n has no solution. (For example, to get 3, take x=-3/2.) - Melvin Peralta, Apr 14 2016

If two independent real random variables, x and y, are distributed according to the same exponential distribution: pdf(x) = lambda * exp(-lambda * x), lambda > 0, then the probability that n-1 <= x/y < n is given by 1/a(n). - Andres Cicuttin, Dec 03 2016

a(n) is equal to the sum of all possible differences between n different pairs of consecutive odd numbers (see example). - Miquel Cerda, Dec 04 2016

a(n+1) is the dimension of the space of vector fields in the plane with polynomial coefficients up to order n. - Martin Licht, Dec 04 2016

It appears that a(n)+3 is the area of the largest possible pond in a square (A268311). - Craig Knecht, May 04 2017

Also the number of 3-cycles in the (n+3)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017